Topics in Calculus — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

(Programme) with Mathematics as non- major / 1. a) Test the continuity at x = 0 of the following function: f(x) = sin* for x * 0, f(0) = 0. b) If u = log(x3 +y*+ z3 - 3xyz), then find the value of *+Yay +2az c) Find the nth differential coefficient of y = sin" x cos x. 2. a) Write the statement of Leibnitz's Theorem, and differentiate n times the following differential equation with respect to x b)Ifu = xF (E) + g (g), then prove that x?. дхz + 2ху ах ду ду2 = 0. c) Discuss the differentiability at x = 0 of the following function: 3. a) Evaluate x cos x - log(1 + x) b) State Lagrange's Mean Value Theorem and use it to show that 1+ x c) State Rolle's Theorem and discuss the applicability of Rolle's Theorem for the function f(x) = |x|, * € l-1,1). 4. a) In the Cauchy's Mean Value Theorem, consider f(x) = 1/x' and g(x) = 1/x in the interval [a, b], then show that c is the harmonic mean of a and b; where a, b, c e c) State Taylor's Theorem with Lagrange's form of remainder. Prove that x"-1 b) Find the double points and points of in flexion, if any, of the curve c) Evaluate cos34xdx. 6. a) By using reduction formula, evaluate b) Trace the curve y (a + x) = x'(3a- x). c) Show that the curve x*-2ax?y-axy?+ay? = 0 has a cusp of the second kind at the origin.